
As pointed out in [1]}, the training of the loss function \(L_{val}^*\)  in (REF )
would require a vast number of samples (often of order millions) to learn accurate approximation of the value function, and is furthermore
prone to overfitting. Indeed, by training a neural network to minimize \(L_{val}^*\) , we would obtain a function which interpolates the random points generated during training. This comes with two shortcomings. First, a large number of training samples is needed to get satisfactory values of the solution and a good enough generalisation to untrained domains. Second, the functions obtained by this method are usually noisy. If we are interested in the derivatives of the PDE solution, as it is the case in finance for example, where greeks are computed in order to hedge contingent claims, the solution computed might not be accurate enough. Some standard methods, such as Ridge and Lasso penalisations allow to reduce overfitting but come at the cost of adding bias and an arbitrary penalty and do not ensure that the derivative of the network will be a good approximation of the derivative of the PDE solution. To circumvent these issues, and following the idea in [1]}, we propose to consider furthermore the learning of the derivative of the value function. This method relies on pathwise differentiation of the target payoff \(\hat{Y}_T^t\)  for deriving the gradient of the value function (see Chapter 7 in [3]}):
\( D_x u(t, \hat{X}_t) &= \; {\mathbb {E}}\Big [\hat{Z}_T^t\big | {\mathcal {F}}_t \Big ], \quad t \in [0,T],\) 
